Elementary POW, May 12-16, 1997

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The Case of the Evaporating Tomatoes
A farmer drives 35 miles to bring 300 pounds of tomatoes to market. Unbeknownst to the farmer, two months ago this village passed a ban on the purchase of tomatoes, and so her entire supply just sits out all day in the sun without anybody buying a single one. During this rather hot day, some of the water in the tomatoes evaporates, so that by the time she packs up her wares to go home, the water has dropped from comprising 99 percent of the total tomato down to constituting 98 percent of the tomato. How much does her supply weigh now? Does the answer surprise you (and if so, why)?
This week's mentor is Steve Earth of Seattle, Washington, who writes:

This was a tricky question. Out of over 60 submissions, there were only 5 correct solvers. Most of the students thought the answer was either 294 or 297, and were not sure why what they did resulted in an incorrect answer. Here is the explanation that I ended up sending to people who were having trouble solving the problem:
  "You're not alone in thinking that's the answer. Many other people replied saying that the tomatoes would weigh 294 or 297 pounds based on the fact that the concentration of tomato went from 1 percent to 2 percent. The flaw in that reasoning is that a percentage change in concentration does not necessarily equate to a percentage change in the whole.
  "A quick example might clarify what I mean here. Consider a bag filled with 2 blue marbles and 3 red marbles. (If you want, imagine that the bag is a tomato with the blue marbles being the water part and the red the solid pulp). In this example, we see that the blue marbles constitute 40 percent of the bag (2 of 5).
  "When 1 blue marble falls out of the bag ("evaporates") - we're left with just 1 blue marble and 3 red. The quantity of blue has dropped by 50 percent (from 2 to 1) and the concentration of blue has dropped by 15 percent (40 percent to 25 percent, 1 of 4). However, none of these four percentages (50 percent, 15 percent, 40 percent, or 25 percent) is the answer to how much the whole amount of marbles has decreased! The overall total in the bag has gone from 5 marbles down to 4 - a decrease of 20 percent (an entirely new number)!
  "So back to our original problem with the tomatoes, we need to do something special with our knowledge that the concentration has gone from 99 to 98 percent - we just can't apply the 99 or 98 percent directly to the total 300 pounds to get the answer..."

Correct Solutions were submitted by:

Gordon Bockus
Mrs Gleichman's 6th Grade Class
Grace Russell Elementary
Wilburton, OK
At first we tried this problem by figuring out how much of the tomatoes was water (.99 x 300), but this didn't help us. So James Bettes asked about how what wouldn't evaporate would stay the same. So we took 1 percent of 300 (.01 x 300 = 3). The new weight would also include these three pounds but it would represent 2 percent of the weight. We set up the following equation: .02 x new = 3 This means that new = 3/.02 = 150. The new weight is 150 pounds. At first this seemed very low because the percentages of water did not change all that much. But if you look at the 3 pound percentages it does make sense. 3 lbs. was 1/100 th of 300 so it was 2/100 of the new weight, and that means the weight would have to have been cut in half.

Greg Schoppe
Homeschooled by mother Clare Schoppe
My sister and I have been doing mixture and solution problems all week, and this is the same kind of problem.
In algebra language:

[Mentor's note: x represents the amount of evaporated water]

(300lbs)(.99) - (x)(1.0) = (300lbs - x)(.98) (300lbs)(.99) - (x)(1.0) = (300lbs)(.98) - (x)(.98) (300lbs)(.99) - (300lbs)(.98) = (x)(1.0) - (x)(.98) (300lbs)(.99 - .98) = (x)(1.0 - .98) (300lbs)(.01) = (x)(.02) 3lbs = 02x 300lbs = 2x 150lbs = x
At first I couldn't believe that half the weight could be lost and only lose 1 percent of the water. I checked the problem to be sure.

(300lbs)(.99) - (150lbs)(1.0) = (300lbs - 150 lbs)(.98)
Even before I finished my check I noticed that the percentage lost was in the same proportion as the total weight lost.

150lbs of water:300lbs of water::1 percent of tomatoes:2 percent of tomatoes

Spencer Rarrick and Dan Reed
9th grade Honors Geometry (teacher: Susan Myers)
Athens Academy
Athens, GA
If 99 percent of the tomatoes is water initially, then 1 percent or 3 lbs. isn't water. After evaporation only 98 percent is water. Since no non-water mass evaporated, the 2 percent left over is still 3 lbs. Therefore the total must be 50(3) or 150 lbs total. The tomatoes lost 1/2 their weight

Bob Coulter
(no school or city given)
We think the answer is 150. You start with 300 lbs. 99 percent water 1 percent pulp. Water is 297 lbs and 3 lbs is pulp. The sun evaporates some water, leaving 98 percent of the weight as water, 2 percent of the weight as pulp. Pulp is still 3 lbs. We have to find what 3 lbs. is 2 percent of. It is 150. (2 percent x 50 = 100 percent, so 3 lbs. x 50 = 150 lbs.)

Tim Peterson
6th grade, Homeschooled
Rochester, NY
My answer is a web page at:

http://pages.prodigy.com/SFSS74C/epow0516.htm
[Mentor's note: Tim also points out on his webpage that a tomato is ordinarily 90 percent water!]

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